Evaluation of PPDs

It is clear from the above discussion that the difference between a PPD and its corresponding determinant solely lies in the coefficients of the permutation P. Unfortunately, this makes PPDs unable to share many of the nice properties of determinants. For instance, the basic multiplicative law valid for determinants 

It can be shown that a PPD of order N may be obtained by evaluating N(N-1 )/2 PPDs of order N-2 as follows, 

With Eq. (42), one expands a PPD by choosing a PPD of order 2 as a minor, and the complementary minor is still a PPD. One can also take a minor from the ASI part. 

Generally, Eq. (45) is much more troublesome than Eq. (42) and it is not essential to the evaluation of a PPD. However, it is of great importance to the application in the VB approach. Using Eq. (45) successively, one can expand a PPD by taking a determinant with order 2 as a minor [39], which is also required in the VB approach. 

The discussion so far is applicable to any given spin number S. It is clear that the expansion of a PPD, Eq. (42), will be greatly simplified if the spin number S= 0. In this case there are not ASI indices any more, and only two values, 1 or -1/2, are taken for du. Furthermore, it is possible to choose a PPD with any even order m as a minor. 

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